Groups and Group Actions: Lecture 13

In which we explore the Orbit-Stabiliser Theorem.

Proposition 58: Let be a group acting on a set . Take , with and lying in the same orbit. Then and are conjugate: there is with . We noted that since and are in the same orbit, there is with . And then we showed that if and only if .

Theorem 59 (Orbit-Stabiliser Theorem): Let be a finite group acting on a set . Take . Then . We defined a map from to and showed that it’s a bijection, then used Lagrange’s theorem.

Corollary 60: Let be a finite group, take . Then , where is the centraliser of in and is the conjugacy class of in . We have already seen that acts on itself via conjugation, and that for we have and , so the result follows immediately from Orbit-Stabiliser.

Proposition 61: Let be prime. Let be a group with order for some . Then the centre of is non-trivial. We used the action of on itself via conjugation. The elements of the centre are precisely the elements whose orbits have size 1. The size of each orbit divides and so is 1 or a multiple of , and since the orbits partition we see that the sum of their sizes is a multiple of . So the number of orbits of size 1 is a multiple of , and since it’s at least 1 it must be at least .

Understanding today’s lecture

By thinking of the group of rotational symmetries of the cube as acting on the edges of the cube, can you show that the group has size 24? What is the size of the group of rotational symmetries of the tetrahedron? Of the dodecahedron?

There’s a hedgehogmaths video in which I talk through the statement and proof of the Orbit-Stabiliser theorem.